The influence of a variable is an important concept in the analysis of Boolean functions. The more general notion of influence of a set of variables on a Boolean function has four separate definitions in the literature. In the present work, we introduce a new definition of influence of a set of variables which is based on the auto-correlation function and develop its basic theory. Among the new results that we obtain are generalisations of the Poincar\'e inequality and the edge expansion property of the influence of a single variable. Further, we obtain new characterisations of resilient and bent functions using the notion of influence. We show that the previous definition of influence due to Fischer et. al. (2002) and Blais (2009) is half the value of the auto-correlation based influence that we introduce. Regarding the other prior notions of influence, we make a detailed study of these and show that each of these definitions do not satisfy one or more desirable properties that a notion of influence may be expected to satisfy.

翻译：变量影响是布尔函数分析中的重要概念。在文献中，一组变量对布尔函数影响的更广义定义存在四种不同表述。本文提出了一种基于自相关函数的新定义，并建立了其基本理论。我们获得的新成果包括泊松不等式与单变量影响的边扩张性质的一般化。进一步利用影响概念，得到了弹性函数和Bent函数的新特征刻画。研究表明，Fischer等人（2002）及Blais（2009）提出的影响定义值仅为本文基于自相关的影响值的一半。针对其他既有影响定义，我们进行了详细分析并证明这些定义均未满足影响力概念应具备的一个或多个理想性质。